J. Phys. Chem. B 2001, 105, 12579-12582
12579
A Novel NMR Approach to Model Percolation in W/O Microemulsions Maura Monduzzi* and Stefania Mele Department Scienze Chimiche, and Consorzio CSGI, Cagliari UniVersity, S.S. 554, BiVio Sestu, 09042 Monserrato Cagliari, Italy ReceiVed: May 30, 2001; In Final Form: October 16, 2001
14 N NMR relaxation rates were measured in ternary DDAB water-in-oil microemulsions to investigate structural transitions along oil dilution lines. A two-step model of relaxation was used to estimate the slow correlation times of the surfactant interface. The interpretation of the 14N slow correlation times in terms of the percolation theory gave critical exponents in agreement with static and dynamic regimes, depending either on the volume fraction of the dispersed phase or on the oil type.
Introduction Modeling of transport properties in microheterogeneous systems such as microemulsions has induced wide debates in the past decade. Particularly, the case of water-in-oil (w/o) microemulsions, which show a percolative behavior when the volume of the dispersed phase Φd or the temperature increase is of considerable interest. Here, to further investigate percolation phenomena, w/o microemulsions formed by the double chain surfactant didodecyl dimethylammonium bromide (DDAB) in the presence of n-decane (DEC), n-dodecane (DOD), and toluene (TOL) oils were chosen. The microstructural features of these systems have been widely investigated by a variety of experimental techniques.1-3 Figure 1 shows the microemulsion (L2 phase) and its nearest liquid crystalline (LC phase) regions of the partial ternary phase diagrams redrawn from previous papers.3,4 In the microemulsion region, the general trend is that along water dilution lines the microstructure is bicontinuous at low water/surfactant (w/s) ratios and disconnected to w/o droplets at high w/s ratios.5 Near the transition point a time dependent water self-diffusion coefficient was measured.6 Similarly, along oil dilution lines, the water continuous network, observed in the proximity of the surfactant/water binary axis, starts to disconnect with increasing oil content.3 Only at very high oil content have w/o spherical droplets been proven to occur for n-alkane systems. In the presence of a high amount of TOL, the self-association of the oil prevents the formation of w/o droplets.7 The microstructural transitions along the water dilution lines have been modeled in terms of the geometrical DOC cylinder model5,6,8-10 with reasonable success. The cubic and lamellar LC phases that appear at low oil content in the diagrams of Figure 1, have bicontinuous microstructure, i.e., oil and water continuous networks coexist, as identified by SAXS, NMR, and optical microscopy.4,11 As a consequence, the microstructure in the neighboring L2 region is expected to be closely related. The microstructural transitions previously suggested along the oil dilution lines3,4 imply a transition from bilayer to monolayer along with a variation of the spontaneous curvature from Ho ≈ 0 to Ho < 0 of the surfactant interface. In this context, Ho < 0 indicates that the * To whom correspondence should be addressed. Phone: 39-070-675 4385; fax: 39-070-675 4388; e-mail:
[email protected] Figure 1. Partial phase diagrams at 25 °C, redrawn from refs 3, 4, 7. The L2 and LC phases are shown together with the oil dilution line at the mass ratio water/surfactant (w/s) ) 0.43 for DEC and DOD systems, and w/s ) 0.16 for TOL system.
curvature of the hydrophobic-hydrophilic interface is convex toward the oil phase, i.e., a water-in-oil state is assumed. The presence of nitrogen in the polar headgroup of the surfactant makes the use of 14N NMR relaxation a sensitive probe to characterize the transitions in terms of interfacial molecular motions. Theoretical Background 14N is a nucleus with a spin quantum number I ) 1, thus the quadrupolar mechanism dominates both the longitudinal (R1) and the transverse (R2) NMR relaxation rates.12 The relaxation data of surfactant systems have been successfully described in terms of the two-step model.13 Within this model, the reduced spectral density function is described by a fast anisotropic motion (τcfast) in the extreme narrowing regime, which is related to the local motions of the observed nucleus, and by a slow isotropic motion (τcslow), which is related to the whole aggregate tumbling:14
(
J(ωN) ) 1 +
)
A2 - S2b J f (0) + S2b J s (ωN) 3
10.1021/jp012042v CCC: $20.00 © 2001 American Chemical Society Published on Web 11/21/2001
(1)
12580 J. Phys. Chem. B, Vol. 105, No. 50, 2001
Monduzzi and Mele
Figure 2. Slow correlation times (τcslow) obtained from 14N NMR relaxation data at 25 °C as a function of the volume fraction of the dispersed phase Φd, along the oil dilution lines at the mass ratio w/s ) 0.43 (b) for DOD, w/s ) 0.43 (O) for DEC, and w/s ) 0.16 (9) for TOL. Data refer to three series of measurements performed on samples prepared by adding different amounts of oil to the water/surfactant mixtures. Vertical bars show the percolation threshold.
where J f(0) is the frequency independent spectral density function of the fast motion and J s(ωN) is the spectral density function associated with the slow motion, ωN is the 14N Larmor frequency, A is the asymmetry parameter of the electric field gradient tensor, and Sb ) 1/2 (3 cos2ϑD -1) is the order parameter, relating the time averaged orientation (ϑD) of the nucleus with respect to the surfactant chain axis. Here, the NMR relaxation measurements have been performed at 7.05 T, at the operating frequency of 21.67 MHz for 14N (extreme narrowing holds well below the value 1/ωN ) 7.34 × 10-9 s). The difference between R2 and R1 relaxation times gives the following relation:
R 2 - R1 )
[ ]
9π2 (S χ)2 (J s(0) + J s (ωN) - 2J s(2ωN)) 40 b
(2)
Here, the product Sbχ can be obtained from the 14N quadrupolar splittings measured in the LR phase, assuming the fraction of the observed nucleus in the bound state Pb ) 1:15 ∆νq ) 3/4 Pbχ Sb. The values of χSb here used are 10.4, 12.8, and 12.675 kHz for DEC, DOD, and TOL systems, respectively. The difference ∆R ) R2 - R1 in eq 2 is mainly affected by slow geometry-dependent motions, since the contribution associated with local fast motions, which are independent of the interfacial geometry, is subtracted.13,14 These slow motions involve surfactant lateral diffusion along the curved interface and reorientation of the interface itself. Thus, the slow correlation time τcslow can be expressed as the sum of the reorientational (τcrot) and diffusional (τcdiff) contributions
(τcslow)-1 ) (τcrot)-1 + (τcdiff)-1
(3)
which for a spherical shape is given as
(τcrot)-1 )
6 Dlat 3kT c -1 (τ ) ) diff 4πηR3H R2D
(4)
Here, η is the viscosity of the dispersing medium, Dlat is the lateral diffusion coefficient of the surfactant, RD and RH are respectively the radius of the water core and the hydrodynamic radius of the spherical w/o droplet. Results and Discussion The 14N NMR relaxation rates were measured along the oil dilution lines shown in Figure 1.3,7 Figure 2 reports the τcslow values, obtained from eq 2 as a function of Φd for DEC, DOD, and TOL systems, and for three different series of samples. It is worth noticing that τcslow gradually increases with increasing Φd, particularly for TOL system where variations over 3 orders of magnitude are observed. Less significant, but still relevant, changes are seen for DEC and DOD systems. In all cases, however, a clear percolation threshold occurs in the proximity of the microemulsion phase boundary. In the LC regions, τcslow increases in the lamellar phase when densely packed bilayers (magnetically and optically anisotropic phase) form at low water content, whereas τcslow decreases in the bicontinuous cubic phase when an isotropic long range order forms at a rather high water content. Evidently, both hydration and local order around the polar head strongly affect the interfacial mobility. Generally when percolation phenomena are considered, macroscopic transport properties of the systems are measured. Conductivity measurements have been found to be rather informative. Suitable interpretative models have related the variation of the transport property to the structural transitions from a disconnected to an interconnected water domain that can occur upon varying Φd or T.16-18 Percolation is proven by a sharp increase of the conductivity, which typically occurs at a critical value Φdc of the volume fraction of the dispersed phase or at a critical temperature Tc (at constant Φd). Then conductivity follows two separate asymptotic scaling power laws having different exponents below and above the percolation threshold.16 As illustrated in eqs 3 and 4, τcslow is a transport property determined by motions such as reorientational tumbling and
Model Percolation in W/O Microemulsions
J. Phys. Chem. B, Vol. 105, No. 50, 2001 12581
diffusion. When w/o droplets interact closely to determine a 3D cluster on the observing time scale, both contributions are expected to increase. In the microemulsion, the slow correlation time τcslow represents the isotropic tumbling of the interfacial aggregates13,14 and depends on the shape and the dimensions of the domain available for the motion. Consequently τcslow may be related to the specific conductivity of charged ions that move inside an interconnected water domain. The introduction of τcslow into the relevant equations of the percolative theory gives
τcslow ) (Φdc - Φd)-s at Φd < Φdc (below percolation threshold) (5) τcslow ) (Φd - Φdc)t at Φd > Φdc (above percolation threshold) (6) In the case of conductivity, the critical exponent t generally ranges between 1.2 and 2, whereas the exponent s depends on the percolation regime, i.e., the observation time scale. The exponent s < 1 (around 0.6) indicates a static percolation regime that is related to the existence of a connected water path in the system (bicontinuous microemulsions).19,20 The exponent s > 1 indicates a dynamic percolation regime21 that is related to the formation of transient water channels due to the merging of droplets during collisions. The use of Φdc ) 0.75 for DOD, Φdc ) 0.62 for DEC, and Φdc ) 0.74 for TOL systems in eqs 5 and 6 results in the loglog linear regression analysis shown in Figure 3. Figure 3a shows three separate regions below the percolation threshold for TOL system. In the range Φd ≈ 0.5-0.7, a critical exponent s ) -0.58 (r ) 0.992), typical of a static percolation regime, is obtained from the linear fitting. The critical values s ) -1.65 (r ) 0.999) and s ) -2.26 (r ) 0.993), calculated in the ranges Φd ≈ 0.2-0.5 and Φd < 0.2 respectively, are typical of a dynamic percolation regime. It may be suggested that transient interactions among the aggregates occur over slightly different time scales. Indeed, for a TOL system at Φd < 0.2, the dimensions of aggregates decrease significantly, as suggested by the 14N NMR relaxation rates R2 very close to R1 values.7 In addition, three different interaction time scales along this oil dilution line of the TOL system may justify the fact that no clear percolation threshold of the conductivity curve was previously observed.4,7 The linear fitting of the data shown in Figure 3b gives the critical exponent t ) 1.16 (r ) 0.992), in agreement with the typical values obtained above the percolation threshold.17 In Figure 3c, the critical exponents s ) -0.6 (r ) 0.993) and s ) -0.57 (r ) 0.995), typical of a static percolation regime, are shown for DEC and DOD systems, respectively. The t critical exponents could not be reliably calculated since a too small region, from above the percolation threshold to the L2 phase boundary, was available. It is remarkable that the critical exponents calculated from NMR relaxation data approach those generally obtained from conductivity measurements. It may be suggested that this is due to the similar time scales over which dynamic phenomena are sampled. Also interesting is that the present results fully agree with previous experimental evidences. NMR quadrupolar relaxation may be suggested as a new powerful and reliable approach at a molecular level to investigate microstructural transitions and percolation phenomena in condensed matter. The noticeable advantage is that the NMR approach, whenever a suitable nucleus is available, can be used even in the absence of ionic conducting species.
Figure 3. Percolative behavior at 25 °C. (a) TOL system below the percolation threshold log τcslow vs -log(Φdc - Φd) - (eq 5) (X) for Φd < 0.2, (9) in the range Φd ≈ 0.2-0.5, (0) in the range Φd ≈ 0.5-0.7; (b) TOL system ([) above the percolation threshold log τcslow vs -log(Φd - Φdc) - eq 6; (c) log τcslow vs -log(Φdc - Φd) - eq 5 for (b) DOD system, and (O) DEC system below the percolation threshold.
Acknowledgment. MURST(Italy), CNR (Italy), and Consorzio Sistemi Grande Interfase (CSGI-Firenze) are acknowledged for support. References and Notes (1) Samseth, J.; Chen, S.-H.; Litster, J. D.; Huang, J. S. J. Appl. Crystallogr. 1988, 21, 835. (2) Eastoe, J.; Heen, R. K. J. Chem. Soc., Faraday Trans. 1994, 90, 487. (3) Monduzzi, M.; Caboi, F.; Larche´, F.; Olsson, U. Langmuir 1997, 13, 2184. (4) Olla, M.; Ambrosone, L.; Monduzzi, M. Colloid Surf. A 1999, 160, 23. (5) Knackstedt, M. A.; Ninham, B. W. Phys. ReV. E 1994, 50, 2839. (6) Knackstedt, M. A.; Ninham, B. W.; Monduzzi, M. Phys. ReV. Lett. 1995, 75, 653. (7) Olla, M.; Monduzzi, M. Langmuir 2000, 16, 6141. (8) Hyde, S. T. J. Phys. Chem. 1989, 93, 1458. (9) Hyde, S. T.; Ninham, B. W.; Zemb, T. J. Phys. Chem. 1989, 93, 1464. (10) Monduzzi, M.; Knackstedt, M. A.; Ninham, B. W. J. Phys. Chem. 1995, 99, 17772.
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